This is a NeoForge version of the mod FM Resource Pack Updater by Raconteur, ported with their permission.
Below is the mod description from the original page, tweaked for the new setup process.

FM Resource Pack Updater

Resource Pack Updater is an extension for the mod Fancy Menu.

It allows you to set a GitHub repository as a source for updating a resource pack.

Fast setup

Github side

You need to have a GitHub repository on which you have the right to create a release. To publish a new version of your resource pack, you just need to publish a new release with your .zip attached to it.

Minecraft side

You need to have FancyMenu and FancyMenu Resource Pack Updater installed. Be careful, FancyMenu depends on other mods to run, you should install them too.

Run Minecraft to generate initial config files - you should see button and text elements added by default to the main menu. Close the game.

Next you'll need to edit /config/rpupdater-common.toml. Put your repository references in the corresponding variables. "rp.name.regex" should be a regular expression corresponding to your RP name. ".*\\.zip" should work with all names possible, but the update will override the first resource pack in your selected list.

Launch Minecraft again. If you did well and the RP is not already set in your game, the update button should appear.

New Features

While porting the mod, I added a few additional features not present in the original version:

By default, the button and text will only show when there is an available update
When updating, the resource pack will forcibly enable itself
If the latest version was already detected but not enabled, it simply enables it
If an older version is present and the update button is pressed, it will remove the old version first
The layout is bundled into the mod itself, removing the need for manual setup. It can be fully customised as any other FancyMenu layout, however renaming the layout may break things.

FancyMenu Resource Pack Updater [NeoForge]